PU-308 · Astrophysics & Cosmology
The unit builds outward from the physics of a single self-gravitating body to the structure and thermal history of the whole universe, showing how gravity, radiation, statistical mechanics, and general relativity combine to determine what stars can exist and how spacetime itself evolves. Students derive the stellar-structure and compact-object limits, then the Friedmann framework, redshift, and thermal history, culminating in a rigorous account of the CMB, nucleosynthesis, inflation, and the growth of cosmic structure.
Lectures
| L01 | Scales, Objects, and Methods of Astrophysics — |
| L02 | Gravitational Energetics and the Virial Theorem |
| L03 | Stellar Structure I: Hydrostatic Equilibrium |
| L04 | Stellar Structure II: Polytropes and Lane-Emden |
| L05 | Radiation, Opacity, and Energy Transport in Stars — |
| L06 | Degenerate Matter and White Dwarfs |
| L07 | The Chandrasekhar Mass and Stellar Endpoints |
| L08 | Relativistic Stars: The TOV Equation |
| L09 | Neutron Stars, Pulsars, and Black Holes |
| L10 | Stellar Atmospheres and the Saha Equation |
| L11 | Galactic Dynamics and Rotation Curves |
| L12 | The Case for Dark Matter |
| L13 | The Cosmological Principle and the FRW Metric |
| L14 | Deriving the Friedmann Equations |
| L15 | Cosmic Fluids and the Acceleration of Expansion |
| L16 | Cosmological Redshift |
| L17 | Distance Measures and Observational Cosmology |
| L18 | Expansion Histories and the Concordance Parameters |
| L19 | The Thermal History of the Universe |
| L20 | The Cosmic Microwave Background |
| L21 | Recombination and Last Scattering |
| L22 | Big Bang Nucleosynthesis |
| L23 | The Horizon and Flatness Problems |
| L24 | Inflation and Slow-Roll Dynamics |
| L25 | Jeans Instability and Gravitational Collapse |
| L26 | Linear Growth of Cosmic Structure |
| L27 | From Perturbations to the Cosmic Web |
| L28 | Synthesis: The Standard Cosmological Model — |
Derivations homed in this unit
Equation of Stellar Hydrostatic Equilibrium
Balancing the local pressure gradient against self-gravity yields dP/dr = -Gm(r)ρ/r² coupled to the mass-continuity equation.
Virial Theorem for Bound Systems
Time-averaging the moment of inertia for a gravitationally bound system gives 2⟨T⟩ + ⟨U⟩ = 0, fixing energetics of stars and clusters.
The Lane-Emden Equation for Polytropes
Assuming a polytropic equation of state P = Kρ^(1+1/n) reduces hydrostatic equilibrium plus Poisson's equation to the dimensionless Lane-Emden equation.
The Chandrasekhar Mass Limit
Combining ultrarelativistic degenerate electron pressure with the n=3 polytrope gives a maximum white-dwarf mass ≈ 1.4 M⊙ independent of radius.
Tolman-Oppenheimer-Volkoff Equation
Solving Einstein's equations for a static isotropic perfect-fluid sphere gives the relativistic generalisation of hydrostatic equilibrium governing neutron stars.
The Saha Ionization Equation
Applying chemical equilibrium of the ionization reaction with Maxwell-Boltzmann and quantum partition functions gives the ratio of ionization states versus temperature and density.
The FRW Metric from Homogeneity and Isotropy
Imposing spatial homogeneity and isotropy on spacetime forces the maximally symmetric Friedmann-Robertson-Walker line element with a single scale factor and curvature constant.
The Friedmann Equations from Einstein's Equations
Feeding the FRW metric and a perfect-fluid stress-energy tensor into the Einstein equations yields the two Friedmann equations plus the cosmic fluid and acceleration equations.
Cosmological Redshift from the Scale Factor
Tracing null geodesics through the FRW metric shows observed wavelength scales with the expansion factor, giving 1+z = a(t_0)/a(t_e).
Comoving, Luminosity, and Angular-Diameter Distances
Integrating the FRW null geodesic and expansion history relates comoving distance to luminosity and angular-diameter distances through factors of (1+z).
Particle Horizon, Horizon and Flatness Problems
Computing the comoving particle horizon and the evolution of the curvature density parameter exposes the horizon and flatness fine-tuning problems of hot big-bang cosmology.
Slow-Roll Inflation from a Scalar Field
A scalar field with dominant potential energy drives quasi-exponential expansion under the slow-roll conditions, solving the horizon and flatness problems.
CMB Blackbody Spectrum and T ∝ 1/a
Adiabatic expansion redshifts a Planck spectrum into another Planck spectrum with temperature falling as the inverse scale factor.
Recombination and Photon Decoupling
Applying the Saha equation to primordial hydrogen fixes the recombination redshift and the last-scattering surface where photons decouple from matter.
Primordial Helium from Big Bang Nucleosynthesis
Neutron-proton freeze-out under weak-interaction equilibrium in the radiation era predicts the primordial helium mass fraction Y ≈ 0.25.
The Jeans Instability Criterion
Perturbing the self-gravitating fluid equations gives a dispersion relation whose instability defines the Jeans length and mass for gravitational collapse.
Linear Growth of Density Perturbations
Linearising the cosmological fluid equations in an expanding background yields the growth equation for δ and its matter- and radiation-era solutions.
Flat Rotation Curves and Dark Matter Halos
Requiring circular-orbit balance for observed flat rotation curves implies an enclosed mass rising linearly with radius, i.e. an isothermal ρ ∝ r⁻² halo.